Questions on the factorial function, $n!=n\times(n-1)\times\cdots\times1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.

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-7
votes
1answer
71 views

What is product of $1!\cdot2!\cdot 3!\cdot…\cdot n!$ [on hold]

Suppose that $F$ is the required function. I need the value of this function till $n$ natural numbers with a direct mathematical expression.
0
votes
2answers
24 views

Simplifying Expression Factorial Expression

I'm confused as how I'm meant to simplify this:$$\frac{(n-2)!}{(n-2-r)!}$$ I have other factorial questions where the variable isn't present in the top factorial like the question above and I'm ...
0
votes
0answers
26 views

Is there a proof that $(n-2x)! \times n^{2x-1} > n!$ (where $x$ is a function related to the prime counting function

Is it possible to prove the following? Let $\pi$ be the prime counting function and $A(n)=\pi(2n)-\pi(n)$ $(n-2A(n))! \times n^{2A(n)-1} > n!$
2
votes
2answers
102 views

Is there any better way to find n! without just multiplying all the numbers till n? [duplicate]

I was solving some permutation, combination problems where we know that we are to use factorial. So I was thinking if there was any shorter way, maybe a formula, than multiplying all the numbers till ...
0
votes
2answers
53 views

Prove or disprove: For non-negative integers $m$ and $n$, $m!n! = (mn)!$

I have rewritten the question as "If $m$ and $n$ are non-negative integers, then $m!n!$ = $(mn)!$" Here is my current attempt. I am not sure if I am on the right path. Proof. Let $m$ and $n$ be ...
2
votes
2answers
38 views

Generalizing Legendre's Theorem on prime factorization of factorials

From Legendre's Theorem we know $$n!=\prod_{p}p^{\lceil \frac{n}{p}\rceil +\lceil \frac{n}{p^{2}}\rceil +.. }$$. Since $\Gamma (n+1)=n!$, i wonder if there is a generalization of this formula for ...
4
votes
2answers
83 views

Last digit of $235!^{69}$

Problem What is the last digit of $235!^{69}$? It's been far too long since I did any modulo calcuations, and even then, the factorial would set me back. My initial thought goes to the last digit ...
3
votes
1answer
48 views

Require assistance proving $n≥2 \Longrightarrow \frac{n!}{n^n} ≤ \frac{1}{2}^{\lfloor \frac{n}{2}\rfloor}$

Theorem: $n≥2 \Longrightarrow \frac{n!}{n^n} ≤ \frac{1}{2}^{\lfloor \frac{n}{2}\rfloor}$ Attempted Solution: We use induction. Additionally, we prove the stronger inequality omitting the floor ...
2
votes
2answers
49 views

Factorial Summation Definition

A while back I found the series $$\sum_{k=0}^n \binom n k (-1)^k (x+k)^n = (-1)^n n!$$ while messing around in Algebra class (specifically when $n$ is any natural number and $x$ is any real number) I ...
0
votes
2answers
38 views

what is the n-k derivative of $x^n$? Also, why is $n!/k! = …$

I am having troubles finding $\frac{d^{n-k}x^n}{dx^{n-k}}$ where $ k \leq n$ I believe it is equal to $n(n-1)(n-2)....k(k+1)x^k$ but htis is just from obersation, I do not know why it's that exactly. ...
0
votes
0answers
22 views

How to find the first and second number of factorial

How to find the first and second number of $40!$ Example $8!=40320$ the first is $4$ and second is $0$ I want to see the solution
5
votes
2answers
139 views

How many numbers of $10$ digits that have at least $5$ different digits are there?

In principle I resolved it as if the first number could be zero, to the end eliminate those that start with zero. The numbers that can use $4$ certain figures (for example, $1$, $2$, $3$ and $4$) are ...
0
votes
2answers
72 views

The method of solving for a factor of $90!$ [duplicate]

If $90! = (90)(89)(88)...(2)(1)$, then what is the exponent of the highest power of $2$ which will divide $90!$ ? How would I apply one of the easiest method from Here? I need help on applying ...
13
votes
0answers
243 views

Factorial of a matrix: what could be the use of it?

Recently on this site, the question was raised how we might define the factorial operation $\mathsf{A}!$ on a square matrix $\mathsf{A}$. The answer, perhaps unsurprisingly, involves the Gamma ...
4
votes
5answers
159 views

Looking for a non-combinatorial proof that $a! \cdot b! \mid (a+b)!$

(I use $a$ and $b$ to denote natural numbers.) Question. Without appealing to the combinatorial interpretation of $$\frac{(a+b)!}{a! b!}$$ as a multinomial coefficients, is there a proof that for ...
2
votes
0answers
29 views

Factorial ratio sum of finite series

Given: $ S = \sum_{i=1}^{n-1}{i! \over n!} $ How would I find the sum for an arbitrarily large $n$ ? Example: $n=5$ $ S = \frac{1!}{5!} + \frac{2!}{5!} + \frac{3!}{5!} + \frac{4!}{5!} = 0.275 $
102
votes
5answers
4k views

How could we define the factorial of a matrix?

Suppose I have a square matrix $\mathsf{A}$ with $\det \mathsf{A}\neq 0$. How could we define the following operation? $$\mathsf{A}!$$ Maybe we could make some simple example, admitted it makes any ...
1
vote
4answers
63 views

Is there a way to evaluate the derivative of $x$! without using Gamma function?

Taking the factorial function $x!$ I wonder if there is a method to find the first derivative of this function without making any use of the Gamma function (or related integral representations of the ...
1
vote
2answers
34 views

$(r-1)^{th}$ derivative of $x^{k+r-1}$

EDIT: added $x^k$ in final answer I want to find: \begin{align} \frac{d^{r-1}}{dx^{r-1}}\left(x^{k+r-1}\right) \end{align} Writing out the first few terms and what I think is the last term we get: ...
5
votes
3answers
518 views

Relationship between factorial and derivatives

I was wondering if there is any relationship between factorials and derivatives because I notice that if we had $x^n$ and we take the $n$-th derivative of this function it will be equal to the ...
2
votes
4answers
78 views

Finding $\lim_{n \to \infty} \dfrac{n^n}{(2n)!}$

Struggling to apply Squeeze THM to find this limit. Specifically, I need a sequence which is always larger than the one in the problem, but which can easily be derived from the middle sequence.
0
votes
2answers
28 views

Factorial with names

Ok so, I have had an argument with my teacher over 1 quiz question that was marked wrong in my data management class. Question. Determine the number of ways that 12 members of the boys' baseball team ...
0
votes
1answer
46 views

Why is 0 factorial 1? [duplicate]

n factorial is product of all numbers between n and 1. 0 factorial is (0 * 1 = 0). Why is 0 factorial 1? How can I proof this in mathematical way?
0
votes
0answers
38 views

Limit of a sequence $a_n = \sqrt[n]{n!}/n$ [duplicate]

Find the limit of the sequence $$a_n = \frac{\sqrt[n]{n!}}{n}$$ I can figure out the limit of the sequence by letting $n=1,2,3,\dots$ but what would be the more conceptual approach to finding the ...
0
votes
0answers
16 views

Solving $(\prod^t_{i=1} N_i m_i)!$

I would like to know how to solve or simplify the factorial $(\prod^t_{i=1} N_i m_i)!$. Here, $i, N_i, m_i$ are positive integers. My effort: $$(\prod^t_{i=1} N_i m_i)!$$ $$\implies (\prod^t_{i=1} ...
3
votes
1answer
90 views

Is $\sum_{p}\frac{1}{p!}=\frac{1}{2!}+\frac{1}{3!}+\frac{1}{5!}+…$ irrational?

Is there known way to determine whether the infinite sum below is rational or not? $$\sum_{p}\frac{1}{p!}=\frac{1}{2!}+\frac{1}{3!}+\frac{1}{5!}+...$$
2
votes
2answers
54 views

Prove $\frac{1\cdot 3\cdots(2n-1)}{2\cdot 4 \cdots(2n)}<\frac{1}{\sqrt{2n+1}}$, $n\ge 1$

Prove $\frac{1\cdot 3\cdots(2n-1)}{2\cdot 4 \cdots(2n)}<\frac{1}{\sqrt{2n+1}}$, $n\ge 1$. I begin by letting $n=1$ then $\frac{1}{2}<\frac{1}{\sqrt{3}}$. Then assume $\frac{1\cdot ...
0
votes
0answers
22 views

Find $\lim_{n \to \infty} \frac{2^n}{n!}$ - Stirling [duplicate]

Find $\lim_{n \to \infty} \frac{2^n}{n!}$. How is possible to solve this limite with Stirling formula? We can solve it with the ratio test, but I asked myself if it's possible with Stirling.
0
votes
1answer
23 views

Factorial grow faster than Exponential - permutation case

It is said that factorial grows faster than exponential, but in the case of permutation: ...
5
votes
1answer
71 views

Summation of factorials.

How do I go about summing this : $$\sum_{r=1}^{n}r\cdot (r+1)!$$ I know how to sum up $r\cdot r!$ But I am not able to do a similar thing with this.
0
votes
0answers
14 views

Solving $\Pi^t_i 2 m_i \left(N_i!\right)^{m_i} $

I would like to work out the result of $\Pi^t_i 2 m_i \left(N_i!\right)^{m_i} $. Here, $t, i, N_i, m_i$ are positive integers. My effort: $$ \Pi^t_i 2 m_i \left(N_i!\right)^{m_i} \implies (2 m_1 ...
2
votes
1answer
38 views

An inequality involving factorials and two variables

The problem is as follows: For $m\ge n>1$ prove that $$(m-2)!(n-1)+(n-2)!(m-1)+(m-2)(n-2)\ge (m-1)(n-1)$$ Since $(m-1)(n-1)-(m-2)(n-2)=m+n-3$ so we only need to show that ...
1
vote
3answers
60 views

How to isolate variable algebraically in a combinatorics equation?

How would I isolate a variable algebraically in a combinatorics equation? For example, if I'm given: $$C(k, 2) = 45$$ How would I solve for $k$, without trying random values of $k$? I know that ...
3
votes
0answers
50 views

Same values for Gamma Function

I was thinking about the Gamma function, which for an integer positive argument is nothing but the factorial function. Using the integral representation, namely $$\Gamma[x] = \int_0^{+\infty}\ ...
0
votes
2answers
93 views

Find all triples of non-negative Integers $a,b,c$ such that $a!b!=a!+b!+c!$ [duplicate]

Exactly what it says in the Title; not much development from there :/
1
vote
2answers
42 views

How to show that for all k, $k! \ge (k/2)^{k/2}$

I'm working on a homework problem that has me showing a "$\Omega(n\log k)$ lower bound on the number of comparisons needed to sort a sequence of $n$ elements, when the input sequence consists of ...
6
votes
1answer
80 views

When $n!=m(m+1)(m+2)$: A Diophantine Equation

I believe that I saw this problem not long ago in a book: Solve the Diophantine Equation $k!=n(n+1)(n+2)$, where $k,n$ are positive integers. However, I am no longer able to find this question, and ...
5
votes
0answers
40 views

Integer solutions for $\lceil{\sqrt{x!}}\rceil^2 - x! = y^2$

Consider the equation $\lceil{\sqrt{x!}}\rceil^2 - x! = y^2$. For $x \le 16$, the equation has the following integer solutions: $$ \begin{matrix} x = 0 & y = 0 \\ x = 1 & y = 0 \\ x = 4 ...
0
votes
1answer
22 views

Perform index shift on summation containing factorial

I'm having a hilariously hard time solving a problem that looks/feels so easy but just won't open up to me. I'm trying to show this equality: $$\sum_{k=0}^{n-1} \frac{1}{k!(n-k-1)!} = \sum_{k=1}^{n} ...
2
votes
3answers
57 views

Problem on factorials and divisiblity of number theory [closed]

How do I prove that $a!b!$ completely divides $(a+b)!$
8
votes
3answers
106 views

Last $500$ digits of $2015!-1$

As the title says, I'm looking for the last $500$ digits of $2015!-1$. I assume it's a repetition of zeroes from the factorial, so the final result is a lot of $9$-s, but I can't formulate a solution ...
0
votes
2answers
490 views

Powerball odds - factorial?

According to Powerball.com, the game is played like this ...we draw five white balls out of a drum with 69 balls and one red ball out of a drum with 26 red balls Their odds explain that the ...
2
votes
8answers
129 views

Why does this series $\sum_{n=0}^{\infty} \frac{(n!)^{2}}{(2n)!}$ converge?

The following series $$\sum_{n=0}^{\infty} \frac{(n!)^{2}}{(2n)!}$$ converges. It fails the divergence test, but once I apply the ratio test, the limit is always equal to $\infty$. Unless you cannot ...
0
votes
1answer
42 views

identity with falling factorials

How can one show that $$\sum_{k=0}^n \frac{(n)_k}{k!} = 2^n$$ for all $n \geq 0$ where for $m \in \mathbb{Z}$ and $k \geq 0$ $(m)_k$ is the "falling factorial": $$(m)_k = \begin{cases} 1, ...
0
votes
1answer
44 views

Fraction Factorial [duplicate]

How do we find factorial of fractions? For eg: $\frac{1!}{2!}=(\frac{\pi}{4})^{\frac{1}{2}}$ Factorials are used in combinatorics and they can only be functioned on integers to give integers.Then how ...
0
votes
1answer
38 views

highest value of 'a'

I got a question when I started factorials Q. If $a^8$ and $8^a$ is completely divisible by $50!$ Then which one of the following is true about 'highest value of a'? (A) $10<a<14$ ...
2
votes
2answers
42 views

Factorial Representation of product

So I've been trying to work out if it is possible to write: $\large \Pi_{i=1}^n (3i-1)$ as an expression involving the quotient or product of two factorials, or really any expression involving ...
4
votes
0answers
43 views

How can I use Stirlings inequality to prove this inequality?

Let $p,k$ be natural numbers with $p\ge k$, show that $$ \frac{(p-k)!}{(p+k)!} \le \frac{1}{p^{2k}} \left(\frac{e}{2} \right)^{2k^2/p}. $$ The text where I come across this says to use Stirling's ...
2
votes
0answers
39 views

Does $y=\lim_{n\to\infty}x\uparrow\uparrow n-f(x,n)$ have a root at $2$?

While messing around on Desmos calculator, I found an interesting factorial/tetration function, where we are using Nutch arrow notation. $$y=\lim_{n\to\infty}x\uparrow\uparrow ...
6
votes
4answers
109 views

Prove: $\forall$ $n\in \mathbb N, (2^n)!$ is divisible by $2^{(2^n)-1}$ and is not divisible by $2^{2^n}$

I assume induction must be used, but I'm having trouble thinking on how to use it when dealing with divisibility when there's no clear, useful way of factorizing the numbers.